Wronskien et équations différentielles $p$-adiques

Abstract

We prove an inequality linking the growth of a generalized Wronskian of $m$ $p$-adic power series to the growth of the ordinary Wronskian of these $m$ power series. A consequence is that if the Wronskian of $m$ entire $p$-adic functions is a non-zero polynomial, then all these functions are polynomials. As an application, we prove that if a linear differential equation with coefficients in $\mathbb C_p[x]$ has a complete system of solutions meromorphic in all $\mathbb C_p$, then all the solutions of the differential equation are rational functions. This is also the case when the linear differential equation has coefficients in $\mathbb Q[x]$, and has, for an infinity of prime numbers $p$, a complete system of meromorphic solutions in a disc of $\mathbb C_p $ with radius strictly greater than $1$.